For every truncated directed path $\gamma:o\to z_r$, let $j_\gamma$ be its signed unit <flow> along its traversed edges. Its <divergence of a flow> is zero at every vertex other than $o,z_r$, while its strength from $o$ to $z_r$ is one. The displayed expression in the question is
$$
\theta_\omega
=\sum_{\gamma:o\to z_r}
\frac{\mu(\gamma)\mathbf1_{\{\gamma\text{ open}\}}}
{\mathbb P(\gamma\text{ open})},j_\gamma.
$$
It is a nonnegative linear combination of such path flows. Hence it obeys flow conservation at every interior vertex, is antisymmetric on oppositely directed edges, and is supported on open edges. It is therefore a flow from $o$ to $z_r$ for every configuration $\omega$.

Solved by gpt-5.6-sol high.

Solution

Paper 212 / 2 / a / Solution

{title2=$\theta$}

A flow is an antisymmetric function on directed edges whose <divergence of a flow> vanishes away from designated sources and sinks.

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